No, it doesn’t solve the problem, but just because the area was initially evangelized from the Persian Empire doesn’t necessarily mean the Chaldeans/ACoE would have exclusive jurisdiction from Persia to the Pacific. That is compounded by the fact that, when the ACoE waned, so too did Christianity east of the Persian Empire (except, of course, in what is now Kerala in India). IIRC, about the only remnant of Syriac Christianity in the Far East is an inscription in Syriac somewhere on the Great Wall of China.
In any case. the area east of the Persian Empire was, at the time, uncharted territory, at least so for ecclesiastical purposes, and hence I think it reasonable to assume that it was fair game, so-to-speak. Rome took advantage of the waning of the ACoE, and established itself in India (and in many ways much to the chagrin of the native Christians in what is now Kerala since they were subjugated, but I digress), and beyond, including China and Japan. As much as I resent Rome’s domination and the claim of the “universality” of the Latin Church, even I will not argue about the lands east of the ancient Persian Empire (except, again, for what is now Kerala), nor the lands in eastern Africa south of Ethiopia. Western Africa (west of Egypt and part of what is now Libya), was Latin Church territory from the get-go. And that brings us to the Americas and Australia, which were not even known to exist much less evangelized.